Pair production of a 125 GeV Higgs boson in MSSM and NMSSM at the ILC
Abstract
In this work we investigate the Higgs pair production in the MSSM and NMSSM at the photonphoton collision of the ILC. We consider various experimental constraints and scan over the parameter space of the MSSM and NMSSM. Then we calculate the cross section of Higgs pair production in the allowed parameter space and compare the results with the predictions in the SM. We find that the large enhancement of the cross section in the MSSM is mainly due to the contributions from the loops mediated by the stau, while in the NMSSM it is mainly due to the contributions from the topsquark loops. For light and large , the production rate can be enhanced by a factor of 18 in the MSSM (relative to the SM prediction). And for a large trilinear soft breaking parameter and a moderately light topsquark, it can also be enhanced by a factor of 2 in the NMSSM. Moreover, we also calculate the values with the LHC Higgs data and display the results for the parameter space with better than the SM value.
pacs:
14.80.Da,14.80.Ly,12.60.JvI Introduction
Since the observation of a resonance near 125 GeV at the LHC in July 2012 1207ATLASCMS (), both the ATLAS and CMS collaborations have accumulated more data and updated their Higgs search results ATLAS13 (); CMS13 (). The combined data corresponding to the integrated luminosities of at 7 TeV and at 8 TeV showed that the observed particle has properties roughly consistent with the Standard Model (SM) Higgs boson. However, the excess in the diphoton signal rate with respect to the SM prediction reported by the ATLAS collaboration may indicate possible physics beyond the SM, such as supersymmetry FebCao (); 1213125GeVHiggs (). So the next important goal of the LHC is to precisely measure its properties, which is also the prime motivation for the future International Linear Collider (ILC) Peskin:2012we (); Ginzburg:2009aa (). With accurate measurement of the Higgs selfcouplings, including the trilinear selfcouplings and the quartic selfcouplings at the ILC, the Higgs potential can be reconstructed properly. And the trilinear Higgs selfcouplings can be measured directly in the Higgs pair production at the ILC through the double Higgsstrahlung process and WW fusion process , which have been investigated comprehensively in eemode (). As an option of the ILC, the photonphoton collider has been paid more attention recently. At the photonphoton collider, the Higgs pair production is oneloop induced process with the new physics contributions and the SM contributions at the same perturbative level, thus the cross section of the Higgs pair production may be enhanced significantly in new physics models. Therefore, the study of Higgs pair production at the photonphoton collider may be a sensitive probe to new physics models.
In the SM the Higgs pair production at a photonphoton collider (i.e.) proceeds through oneloop triangle and box diagrams induced by the heavy quarks and bosons Jikia:1992mt (). This process may also be a sensitive probe for new physics and has been studied in various new physics models, such as 2HDM Zhu:1997nz (); Cornet:2008nq (), the vectorlike quark model Asakawa:2010xj () and the supersymmetric (SUSY) models Zhou:2003ss (). The cross section of in these new physics models can deviate significantly from the SM prediction due to the oneloop correction to the trilinear Higgs selfcouplings Asakawa:2010xj (). For the Higgs boson near 125 GeV, the most efficient detectable channel at the photonphoton collider is . The backgrounds, such as , and , can be suppressed if correct assignment of tracks to parent partons and appropriate invariantmass and angular cut are achieved Kawada:2012uy (); Belusevic:2004pz ().
With the updated experimental data at the LHC, the constraints on the parameter space of SUSY models have been becoming more and more stringent. Therefore, motivated by the latest experimental results, we assume a SMlike Higgs boson in 123127 GeV and study its pair production at the photonphoton collider in the Minimal Supersymmetric Standard Model (MSSM) MSSM (). In the MSSM, the pair production of the SMlike Higgs boson receives additional contributions from loops mediated by the third generation squarks, staus, the charginos, and also the charged Higgs bosons. It was found that the cross section of can be enhanced due to the nondecoupling effect and the mixing between the lefthanded and righthanded stau. In the NexttoMinimal Supersymmetric Standard Model (NMSSM) NMSSM (), which is more favored by the experimental data JulyCao (), the studies of are still absent. Therefore, it is necessary to investigate the SMlike Higgs pair production in the NMSSM and compare the predictions with the MSSM results.
This work is organized as follows. We first briefly describe the features of the Higgs sector in the MSSM and NMSSM in Section II. Then we present our numerical results for the Higgs pair production in both models in Section III. Finally, we draw our conclusions in Section IV.
Ii Higgs sector in MSSM and NMSSM
As the most economical realization of supersymmetry in particle physics, the MSSM consists of two Higgs doublet and . Its superpotential has the form as follows,
(1) 
where , and denoting the squark superfields, and denoting slepton superfields, and () being the corresponding Yukawa coupling coefficients. After the electroweak symmetry breaking, the MSSM predicts two physical CPeven Higgs bosons and , one physical CPodd Higgs boson and a pair of charged Higgs bosons . At tree level, this Higgs sector is determined by the mass of the CPodd Higgs and the ratio of the Higgs vacuum expectation values . In most cases of the MSSM, the lightest Higgs boson is SMlike (with the largest coupling to vector bosons), and for large and moderate , the mass is given byCarenaHiggsmass ()
(2) 
with GeV, ( and denote the stop masses), ( denotes the trilinear Higgsstop coupling). Obviously, in order to lift the Higgs boson mass up to about 125 GeV, large or is needed, which in turn usually requires a large .
Since the MSSM suffers from problem and large and induce some extent of finetuning, the NMSSM has been intensively studied. Its superpotential is given by
(3) 
with being singlet Higgs superfield and dimensionless parameters and denoting the coupling strengths of Higgs selfinteractions. Note that when the singlet field develops a vacuum expectation value , an effective term is generated by . Compared with the MSSM, the NMSSM predicts one more CPeven Higgs boson and one more CPodd Higgs boson.
Due to the coupling in the superpotential, there is additional tree level contribution to the SMlike Higgs boson mass, i.e. . Moreover, the mixing between the singlet and doublet Higgs fields can significantly alter the Higgs boson mass. Affected by the above two factors, for and , the mass of the SMlike Higgs boson can reach 125 GeV even without the radiative correction, which can significantly ameliorate the finetuning suffered by the MSSM.
In the limit and is fixed, the singlet field decouples from the doublet Higgs sector so that the NMSSM phenomenology reduces to the MSSM. So in order to compare the Higgs sector between the two models, we require in the NMSSM and consider two scenarios:

NMSSM1 scenario: The lightest Higgs boson acts as the SMlike Higgs boson . In this scenario, the mixing effect is to pull down , and if the mixing effect is dominant, large radiative correction is needed to predict .

NMSSM2 scenario: The nexttolightest Higgs boson acts as the SMlike Higgs boson . In this scenario, the mixing effect is to push up . Both the mixing effect and the additional tree level contribution make the large radiative correction unnecessary.
Iii Calculations and numerical results
At the ILC, the photonphoton collider can be achieved from Compton backscattering of laser photon off beams, so the total cross section of can be obtained in the form
(4) 
where () is the center of mass energies of (), and is distribution function of photon luminosity, which is defined as
(5) 
For the unpolarized initial electrons and laser photon beams, the energy spectrum of the backscattered photon is given byphotonmode ()
(6) 
In SUSY models the SMlike Higgs pair production at the photonphoton collider receives additional contributions from the loops mediated by the third generation squarks, staus, charginos and the charged Higgs boson, which are at the same perturbative order with the SM contributions. Therefore, the cross section of the Higgs pair production in SUSY may be enhanced significantly with respect to that in the SM. In the calculations, we adopt the t Hooft Feynman gauge 'tHooft:1971rn (), which involves a large number of diagrams from the loops mediated by W bosons, charged Higgs bosons, the Goldstone particles and also the ghost particles, so we do not present the Feynman diagrams from these particles, which can be seen in Zhu:1997nz (); Zhou:2003ss (), and label these diagrams as the socalled WC diagrams. We only show the Feynman diagrams arising from the third generation squarks, staus and charginos in Fig. 1, which can be divided into six parts labeled from (i) to (vi), and each part is UV finite and gauge invariant. The calculations of the loop diagrams are usually tedious if one expands the tensor loop functions in terms of scalar loop functions. So we retain the tensor loop functions and use the improved LoopToolslooptools () to calculate them. In Fortran code, we use arrays to encode the tensor loop functions as well as other quantities such as Lorentz vectors, Dirac spinors and Dirac matricesCao:2007dk (). The analytical expressions are so lengthy that we do not presented explicit forms here.
In the numerical calculations we take GeV, GeV, =1.78 GeV GeV and PDG (), and fix the center of mass energy of ILC to be 1 TeV. For GeV, the cross section of in the SM is 0.63 fb, which changes little for varies from 123 GeV to 127 GeV. And we also numerically checked our results, which are agreement with Jikia:1992mt ().
In this work, we use the package NMSSMTools NMSSMTools () to scan over the parameter space of the MSSM and NMSSM, and investigate the samples which predict a SMlike Higgs boson within GeV. The scan ranges of the parameter space are same as JulyCao (), and the surviving samples satisfy the following experimental constraints: (1)the LHC constraints on the nonstandard Higgs boson and the mass of sparticles; (2) the limits from the muon anomalous magnetic moment, the electroweak precision data and various Bphysics observables, such as the latest experimental result of Bsmumu (); (3) the constraints from dark matter relic density ( range given by the WMAP) as well as the direct search result from XENON2012 experiment (at 90% confidence level); (4) the global fit of the SUSY predictions on various Higgs signals to the LHC Higgs data Giardino:2012dp (); Espinosa:2012ir (); Ellis:2013lra (); Giardino:2013bma (). For each surviving samples, we calculate the Higgs pair production rate , which is less sensitive to higher order QCD corrections.
In Fig. 2 we project the surviving samples of the MSSM and NMSSM in the plane of versus the mass of SMlike Higgs boson. For the case in the NMSSM, we show the results for the NMSSM1 and NMSSM2 scenario separately. The figure shows that in most cases the cross sections of Higgs pair production are slightly enhanced in the MSSM and NMSSM with respect to that in the SM. However, in some special cases, the normalized production rate can reach 18 in the MSSM and 2 in the NMSSM. From the figure we can also see that the production rates in the NMSSM1 scenario usually slightly larger than that in the NMSSM2 scenario, and in the NMSSM2 scenario the production rates can also be suppressed. The reasons will be explained later.
Compared the normalized production rate of process at the ILC with the process at the LHCCao:2013si (), we find the normalized production rate of can be much larger than that of in the MSSM, while it is usually relatively smaller in the NMSSM. In the MSSM, it is the contributions from the stau loops, which are absent for the process , that may be large enough to enhance the normalized production rate of significantly. While in the NMSSM, it is because the negligible contributions from the stau loops and the destructive interference between the contributions from the squark loops and W boson loops in the process , in which the contributions from the W boson loops may be dominant. From the numerical calculations we also find that the contributions from the chargino loops and bottomsquark loops to the cross section of are quite small. And the large enhancement of the cross section in the MSSM is mainly due to the contributions from the stau loops, while in the NMSSM it is mainly due to the contributions from the topsquark loops, which is similar with the process at the LHC.
For the large normalized production rate in the MSSM, its main SUSY contributions come from the stau loops with chiral flipping (i.e. the diagrams (i) and (ii) in Fig.1). For light , the amplitudes of these diagrams scale like . So for light and large , the normalized production rate in the MSSM can be enhanced significantly, which can be seen clearly in Fig.3. Compared the contributions from the stau loops, the contributions from topsquark loops in the MSSM are relatively small. However, it can still enhance the cross section slightly. While in the NMSSM, due to the small , the contributions from the stau loops are negligible and the effect of topsquark loops is remarkable. Therefore, in Fig. 4 and Fig.5 we show the normalized production rate as a function of and , respectively.
From the Fig.4 and Fig.5 we can see that large and light usually predict a large normalized production rate in the NMSSM. This is because for a light , the dominant contributions from topsquark loops scale like Cao:2013si (). Fig.5 also indicates that in a corner of the parameter space in the NMSSM1 scenario, the deviation can be larger than 40% for 1.5 TeV. In such cases, the contributions from the socalled WC diagrams are usually larger than the cases in the NMSSM2 scenario. In fact, it is the contributions from the charged Higgs boson loops and Goldstone boson loops that lead to the larger enhancement in the NMSSM1 scenario. Different from the couplings in the SM, the coupling of SMlike Higgs boson with charged Higgs boson or Goldstone boson in the NMSSM has additional interactions from the Higgs singlet field Franke:1995tc (),
(7) 
where denotes the singlet component of the SMlike Higgs boson. Note that NMSSM1 scenario usually prefers larger and smaller than NMSSM2 scenarioFebCao (). And for these samples, we have numerically checked that the couplings and are usually larger in the NMSSM1 scenario than that in the NMSSM2 scenario.
Note that even for heavy , the normalized production rate in the MSSM can also be enhanced slightly. This is mainly due to the contributions from topsquark loops. For heavy , the amplitude of the dominant diagrams from topsquark loops can be written as
(8) 
with denotes the top quark Yukawa coupling, is the chiral mixing angle and and are (1) coefficient with opposite signs. In this case, the mass splitting between and is small (i.e. ), so the two terms in Eq.(8) cancel severely. However, because is usually larger than stop mass, the cross section of Higgs pair production in the MSSM can still be enhanced by about 10%.
As analyzed in the section II, in order to predict a 125 GeV Higgs in the MSSM, the average stop mass or the trilinear soft breaking parameter must be large. However, in the NMSSM1 scenario, the mixing effect is destructive with the additional tree level contribution, so large radiative corrections are needed to predict a 125 GeV Higgs boson. For the same values of the topsquark in the NMSSM, the NMSSM1 scenario usually prefers a larger , which leads to the larger production rate than the NMSSM2 scenario, as shown in Fig. 2. In the NMSSM2 scenario, both the additional tree level contribution and the mixing effect can enhance the Higgs boson mass up to about 125 GeV, so the constraints on the parameter is not so strong. That is why the production rate can also be enhanced or suppressed in the NMSSM2 scenario.
Finally, we calculate the values with the LHC Higgs data for the samples with 125 GeV126 GeV in the best fitted mass region Giardino:2012dp (); Espinosa:2012ir (); Ellis:2013lra (); Giardino:2013bma (), and show the relationship with the normalized production rate in Fig. 6. From the figure we can see that there exist some samples with slightly smaller than its SM value (). For the better than the SM value, the production rates only deviate slightly from the SM prediction in the NMSSM2 scenario, while the deviation can reach 40 in the NMSSM1 scenario and the production rate in the MSSM can be enhanced about 18 times larger than the SM prediction. And we also numerically check that, for the samples with much larger than its SM value, the coupling of Higgs boson to b quark usually deviates significantly from its SM prediction, or/and the singlet component of the SMlike Higgs boson is usually large.
Iv Summary and Conclusion
For the precise measurement of the Higgs boson properties at the ILC, the Higgs pair production at the photonphoton collision will play an important role since it can test the Higgs selfcoupling. In this work we investigated the Higgs pair production in the MSSM and NMSSM at the photonphoton collision of the ILC. We considered various experimental constraints and scanned over the parameter space of the MSSM and NMSSM. Then in the allowed parameter space we calculated the cross section of Higgs pair production at the ILC and compared the results with the predictions in the SM. We found that the large enhancement of the cross section in the MSSM is mainly due to the contributions from the loops mediated by the stau, while in the NMSSM it is mainly due to the contributions from the topsquark loops. For light and large , the normalized production rate in the MSSM can reach 18. And for a large trilinear soft breaking parameter and a moderate topsquark mass , the normalized production rate can also reach 2 in the NMSSM. We also calculated the values with the LHC Higgs data. For the better than the SM value, the production rates only deviate slightly from the SM prediction in the NMSSM2 scenario, while the deviation can reach 40 in the NMSSM1 scenario and the production rates in the MSSM can be enhanced about 18 times larger than the SM prediction.
Acknowledgement
We thank Jin Min Yang, Junjie Cao and Jingya Zhu for helpful discussions. This work was supported in part by the National Natural Science Foundation of China (NNSFC) under grant No. 11247268.
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